Deep material networks for fiber suspensions with infinite material contrast

TL;DR

This work advances the homogenization of fiber suspensions in non-Newtonian solvents by introducing the Flexible DMN (FDMN), a deep-network framework that can handle incompressible fluids and infinite material contrast. It derives two-phase layered-emulsion blocks and coated layered materials (CLMs) as non-singular, physics-based building blocks, and embeds them into a DMN-like architecture to predict non-linear, shear-thinning responses with high accuracy. The approach achieves validation errors below 4.31% across 31 fiber orientations, six load cases, and a broad shear-rate range, while delivering substantial speedups (up to ~1.7e4×) over FFT-based homogenization after offline training. Compared to prior ML-aided analytical models, the FDMN provides greater flexibility and consistency with the underlying physics, at the cost of increased training data and computational effort for data generation, enabling more generalizable concurrent two-scale simulations in polymer composites and related systems.

Abstract

We extend the laminate based framework of direct Deep Material Networks (DMNs) to treat suspensions of rigid fibers in a non-Newtonian solvent. To do so, we derive two-phase homogenization blocks that are capable of treating incompressible fluid phases and infinite material contrast. In particular, we leverage existing results for linear elastic laminates to identify closed form expressions for the linear homogenization functions of two-phase layered emulsions. To treat infinite material contrast, we rely on the repeated layering of two-phase layered emulsions in the form of coated layered materials. We derive necessary and sufficient conditions which ensure that the effective properties of coated layered materials with incompressible phases are non-singular, even if one of the phases is rigid. With the derived homogenization blocks and non-singularity conditions at hand, we present a novel DMN architecture, which we name the Flexible DMN (FDMN) architecture. We build and train FDMNs to predict the effective stress response of shear-thinning fiber suspensions with a Cross-type matrix material. For 31 fiber orientation states, six load cases, and over a wide range of shear rates relevant to engineering processes, the FDMNs achieve validation errors below 4.31% when compared to direct numerical simulations with Fast-Fourier-Transform based computational techniques. Compared to a conventional machine learning approach introduced previously by the consortium of authors, FDMNs offer better accuracy at an increased computational cost for the considered material and flow scenarios.

Figures (8)

• Figure 1: Rank-three, coated layered materials with orthogonal layering directions (a), and non-orthogonal layering directions (b). The core material is shown in red and the coating material is shown in gray. We hint at the separation of length scales between the material layers through our choice of the layer thicknesses.
• Figure 2: Weight and stiffness propagation (from the bottom to the top) in a two-phase direct DMN gajek2021fe of depth K = 3, by Gajek et al. gajek2022fe, licensed under https://creativecommons.org/licenses/by/4.0
• Figure 3: Weight and material property propagation (from the bottom to the top) in a two-phase FDMN gajek2021fe of depth $K=3$ with rank-$3$ CLMs
• Figure 4: Weight and material property propagation (from the bottom to the top) in a two-phase DMN of depth $K=3$, representing a two-phase rank-$3$ CLM
• Figure 5: Fiber orientation triangle $S_{\sf{T}}$ in CMYK coloring with 31 evaluation points (a), and material data with Cross-type fit for Ultramid®B3K (b).